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Tag 0BK3

Chapter 48: Local Cohomology > Section 48.4: Finiteness of pushforwards, I

Proposition 48.4.7 (Kollár). Let $j : U \to X$ be an open immersion of locally Noetherian schemes with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module. The following are equivalent

  1. $j_*\mathcal{F}$ is coherent,
  2. for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{x\}}$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{\overline{\{x\}}, z}^\wedge$ we have $\dim(\mathcal{O}_{\overline{\{x\}}, z}^\wedge/\mathfrak p) \geq 2$.

Proof. If (2) holds we get (1) by a combination of Lemmas 48.4.1, Remark 48.4.5, and Lemma 48.4.4. If (2) does not hold, then $j_*i_{x, *}\mathcal{O}_{W_x}$ is not finite for some $x \in \text{Ass}(\mathcal{F})$ by the discussion in Remark 48.4.6 (and Remark 48.4.5). Thus $j_*\mathcal{F}$ is not coherent by Lemma 48.4.1. $\square$

    The code snippet corresponding to this tag is a part of the file local-cohomology.tex and is located in lines 796–811 (see updates for more information).

    \begin{proposition}[Koll\'ar]
    \label{proposition-kollar}
    \begin{reference}
    Theorem of Koll\'ar stated in an email dated Wed, 1 Jul 2015.
    \end{reference}
    Let $j : U \to X$ be an open immersion of locally Noetherian schemes
    with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module.
    The following are equivalent
    \begin{enumerate}
    \item $j_*\mathcal{F}$ is coherent,
    \item for $x \in \text{Ass}(\mathcal{F})$ and
    $z \in Z \cap \overline{\{x\}}$ and any associated prime
    $\mathfrak p$ of the completion $\mathcal{O}_{\overline{\{x\}}, z}^\wedge$
    we have $\dim(\mathcal{O}_{\overline{\{x\}}, z}^\wedge/\mathfrak p) \geq 2$.
    \end{enumerate}
    \end{proposition}
    
    \begin{proof}
    If (2) holds we get (1) by a combination of
    Lemmas \ref{lemma-check-finiteness-pushforward-on-associated-points},
    Remark \ref{remark-closure}, and
    Lemma \ref{lemma-sharp-finiteness-pushforward}.
    If (2) does not hold, then $j_*i_{x, *}\mathcal{O}_{W_x}$ is not finite
    for some $x \in \text{Ass}(\mathcal{F})$ by the discussion in
    Remark \ref{remark-no-finiteness-pushforward}
    (and Remark \ref{remark-closure}).
    Thus $j_*\mathcal{F}$ is not coherent by
    Lemma \ref{lemma-check-finiteness-pushforward-on-associated-points}.
    \end{proof}

    References

    Theorem of Kollár stated in an email dated Wed, 1 Jul 2015.

    Comments (1)

    Comment #2598 by Rogier Brussee on June 5, 2017 a 10:39 am UTC

    Suggested slogan: (weak analogue of Hartogh's theorem) On Noetherian schemes, the restriction of a coherent sheaf to an open set with complement of codimension 2 in the sheaf's support, is coherent.

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